From Subjectivity to Polycontextural Logic

Johannes Falk on April 10, 2026 · 17 min read

Subjectivity means different observers reach different yet valid conclusions. Gotthard Günther’s polycontextural logic turns that into a formal account of self-reflection, and shows why today’s mono-contextural AI cannot manage it. Two network models of consensus and graph colouring make the idea concrete.

Subjectivity in philosophy is the property that distinguishes one subject from another, with these differences manifesting in desires, opinions, tastes, and dispositions. Every aspect of our daily lives is shaped by the subjectivity of those involved: to serve the different tastes of all listeners, a radio station plays music from various genres. Due to differing political views, politicians debate and group themselves into parties. Because of inherent subjectivity, it is impossible for individual subjects to provide an independent, and thus objective, assessment of observations: images that one person regards as beautiful or aesthetic may be seen by another as kitsch. There is no arguing about taste — it is subjective (see also the Figure).

Through norms and definitions, it is possible to formulate statements that can then be evaluated objectively, i.e. according to the corresponding definition. Based on the axioms of mathematics, we agree, for example, that (in classical mathematics) 5 > 2 holds. This observation already reflects the fact that mathematics and logic consider themselves objective, intersubjective sciences. This implies that their definitional foundation consists of non-derivable axioms, which must be taken as given and cannot be proved. It is therefore not surprising that over the course of the last century, inconsistencies were discovered at various points in mathematics, which had to be resolved by additional axiomatic restrictions (e.g. Zermelo-Fraenkel set theory (ZF), which was framed so narrowly that antinomies — contradictions — from Cantor’s set theory are no longer possible). Even though no antinomy has yet been found in ZF, the actual freedom from antinomies remains unprovable.

For many years, a subfield of computer science has been gaining increasing attention: Artificial Intelligence (AI). The goal of AI is for machines to approach human behaviour and decisions in their own behaviour and decisions. Modelled on the way humans learn, AI is trained using large amounts of pre-defined training data. Recalling the previous two paragraphs, we observe an interesting contradiction: the algorithms of AI are based on logical processes designed according to the laws of mathematics and logic. Yet the intended goal of these algorithms is to replicate the capabilities of the human brain — that is, the capabilities of a system that clearly does not work objectively, but rather — each for itself — possesses its own subjectivity.

The philosopher Gotthard Günther engaged intensively with the role of subjectivity in his work (Günther, 1976, 1979). Building on Hegel’s dialectic, he postulated that every knowing subject spans its own logical system: a contexture. Within its own contexture, the rules of classical logic apply. However, the fundamental notions of “true” and “false” can differ across contextures, which is what guarantees subjectivity. In the interplay of different contextures, situations can arise in which the same state of affairs produces two different logical answers.

According to Günther, by splitting an all-encompassing logic into various interconnected contextures — the so-called polycontextural logic (PCL) — a human capacity can be explained that has so far remained inaccessible to any AI: we can reflect on our actions and our place in the world; we question our behaviour from an externally conceived perspective.

In classical mathematical logic, self-references, and thus self-reflection, inevitably lead to logical problems. For this very reason, ZF axiomatically excludes sets that contain themselves (axiom of foundation). In Günther’s polycontextural logic, such self-references can be resolved by distributing the contradictions across different contextures. The interaction between the various contextures can then, according to Günther, be represented via a proömial relation.

Self-Reflection in PCL: The Proömial Relation

The proömial relation is what Günther regards as the fundamental building block of self-reflection. To make it concrete, we consider three subjects (A, B, C). These are mono-contextural, meaning that each of these subjects possesses only one (namely its own) logic. Each subject therefore acts in a maximally egocentric manner and is cognitively unable to de-objectify its own subjective perspective. The proömial relation is depicted graphically in the second figure.

We begin with an object O (represented by a clock in the figure), which is observed by a subject (A). Upon observing it, the subject forms an opinion about the object — for example, about the passage of time. Similarly, a second subject (B) also observes the clock. Due to their different subjectivities, it is possible that A and B form different opinions about O. A mutual exchange of views will lead to no agreement, since, owing to each subject’s closed internal logic, each insists on its own opinion.

Now, alongside A and B, we introduce a third subject C. This subject observes the observation processes of A and B. That is, C does not form an opinion about O, but rather about the manner in which A and B arrive at their opinions. C may arrive at two different conclusions: either C believes that at least one of the two subjects is forming its opinion incorrectly, or C believes that both observed subjects are observing correctly.

From C’s perspective, however, this creates a logically problematic situation: A and B both observe correctly, yet arrive at different results. The only possible conclusion C can draw from this observation — and C’s own egocentrism plays no role here — is: both A and B are right. What C can learn from this for itself is that, evidently — depending on one’s logical standpoint — different and yet jointly correct opinions can exist.

For A and B, this insight naturally remains something they reject. It contradicts their egocentric worldview that their own logic is not identical to a globally valid logic.

In the process described, C learns that the correct evaluation of a state of affairs can differ depending on one’s logical standpoint. Above all with regard to itself, this insight questions C’s own egocentrism, or in philosophical terms: it enables self-reflection.

Excursus: Transformations in Physics

The processes just sketched can be well illustrated by an example from quantum mechanics (Falk et al., 2021). Wave–particle duality states that light is simultaneously both a wave and a particle. Depending on the measuring apparatus used, however, only one of these properties can be measured:

• In the double-slit experiment, interference patterns can be observed on the observation screen, indicating constructive and destructive superposition of light waves. By measuring the resulting patterns, the exact wavelength can even be calculated.
• In the analysis of the photoelectric effect, the emission of electrons can be observed when a metal plate is irradiated with light. Interestingly, the energy of the emitted electrons does not increase with greater intensity of the incident light. Instead, more electrons are emitted. This behaviour can only be explained by assuming that light consists of small particles, so that greater intensity means a higher density of particles.

Analogous to the arrangement of different subjects described above and depicted in Figure 2, wave–particle duality can be interpreted as a polycontextural problem. We assume that subjects A and B are physicists, both investigating a light source (O). A conducts a double-slit experiment; B conducts a photoelectric effect experiment. Consequently, A becomes convinced that light is composed of waves, while B believes that light consists of particles. Each physicist knows that they have conducted their experiment entirely correctly. From their limited perspective, each must therefore conclude that the other physicist’s conviction is wrong. A third subject C can now observe the two active physicists without preconception. C forms no opinion about the results, but merely observes that both physicists are conducting their experiments correctly. For C, the only logical conclusion remains: both are right. C thus arrives at the insight that light is both wave and particle.

The Logical Boundary of Modern AI

The AI systems known to us belong to the category of weak AI. This means that while they are capable of addressing specific problems, they cannot — as a human might — independently identify tasks and independently acquire the corresponding knowledge. In contrast to weak AI, a strong AI would not act exclusively reactively, but would also proactively question, optimise, or discontinue its own actions. Why does strong AI not yet exist? Based on the discussion in the previous chapter, this can be answered: all existing AI approaches are built on the familiar classical assumption that a universal logic with globally defined truth values should hold everywhere. Any AI defined in this way is therefore necessarily mono-contextural and consequently incapable of de-objectifying its own subjective perspective. This immediately renders a central capability of strong AI impossible: self-reflection.

A modern approach to implementing strong AI would therefore need to shed the straitjacket of global logic and allow each acting instance to judge right and wrong for itself. Intuitively, however, this leads to a problem: how can a set of such instances organise themselves and perform meaningful actions when each instance can decide for itself whether an action is meaningful? A coarse goal is therefore needed, towards which all instances strive. In nature, such fundamental goals are indeed known — for example as instincts, i.e. innate inner foundations for certain behaviours. The question remains, however, whether — and if so how — different instances that follow the same basic instincts but ultimately operate on different logical foundations can interact with one another. To answer this question experimentally (i.e. through numerical simulations), we consider the following model (Falk et al., 2022):

We consider a large number of different agents (logical instances) connected to one another in the form of a network. Each agent is equipped with an observable attribute — in our model, a colour. Importantly, the colour stands in for attributes of any kind, including opinions or convictions. As the simplest variant of an instinct, each agent strives for conformity — that is, agreement with the observed society, its network neighbours. Each agent thus wishes to adapt its own colour to that of its neighbours. From a social-psychological perspective, such conformity can be motivated by an inner need for a sense of belonging.

In accordance with our above-stated requirement for a polycontextural world, each agent is to operate with a different logic. We generate this in the model by assuming that each agent perceives colours differently. Each instance is equipped with a translation table that maps the external colour (the pseudo-objective logic) to the internal colour (the subjective logic) (Figure 3).

Randomly, an agent is selected, observes the colour of a direct neighbour (according to its own translation table), and then colours itself in the colour it subjectively understands as conforming. Over time, no uniform colouring will emerge — no agreement on true/false, or global logic, emerges. The obvious reason is that each agent acts mono-contextually (analogously to subjects A and B in the proömial relation).

To enable a simple form of self-reflection, we extend the model by one component: each agent records how often the observation of a neighbour has led to a change in its own colour. If a globally valid logic existed, neighbouring agents would quickly agree on a colour and colour changes would cease. The frequency of colour changes therefore indirectly reflects how well the agent’s own logic and the logic of the neighbours — who are themselves observers — are compatible. If an instance has to change colour too frequently, it concludes that its own logic is not in conformity with its environment and generates a new logic for itself (a new colour table).

With this model, we can observe that the instances gradually agree on a common logic over time. We can therefore demonstrate that different instances, initially based on different logics, can synchronise through simple rules such that a common logic emerges.

Filter Bubbles and Polarisation

As sketched above, networked agents can agree on a common logic through communication. With reference to social media, however, we have increasingly observed in recent years the phenomenon of filter bubbles and opinion polarisation. This involves the coexistence of groups holding incompatible opinions on certain topics. These filter bubbles might also be interpreted as groups of individuals sharing a common logic, which is however incompatible with the logic of the other group. This contradicts our above observation that networked agents converge on a common logic over time. An interesting question is therefore what influence the structure of the network has on the global synchronisation of local logics.

In principle, such an analysis would be possible with the model described above. To enable better comparability with other approaches, however, we also draw on a further model. This model — graph colouring dynamics — uses a different form of ‘instinct’. Instead of conformity with the environment, each instance attempts to achieve maximum individuality. Application areas of the model explained below include in particular conflict-free resource planning by distributed systems (Hadzhiev et al., 2009).

We begin the explanation of this model with three definitions from graph theory:

1. A graph is said to be coloured if a colour is assigned to each node.
2. A graph is validly coloured if no two adjacent nodes share the same colour.
3. For every graph, there exists a number k (the so-called chromatic number) of colours that are minimally required so that the graph can be validly coloured (see also Figure 4).

In the context of graph colouring, there are two major problems: first, determining the chromatic number of a given graph, and second, actually colouring a graph for a given chromatic number. Both problems are presumably not efficiently solvable (NP-hard and NP-complete). In the model considered here, we will not address these open questions further. Instead, we restrict ourselves to very simple graphs for which the chromatic number and the correct colouring can be immediately recognised: ring graphs with an even number of nodes. These graphs can trivially be coloured in the pattern ABAB, and it is immediately clear that the chromatic number is k=2.

Conventional colouring algorithms would find this solution without much effort: given the limited number of colours, an algorithm would simply start from an initial node and colour it in one of the two possible colours. The neighbouring nodes must then necessarily receive the second colour, and their neighbours in turn the first colour again. The correct solution therefore propagates in a wave from the initial node across the entire network.

In the course of our analysis of polycontextural systems, we wish to take a different approach here. Each node is in our case its own logical instance, possessing information only about its immediate surroundings. Based on this information, the individual nodes are to colour themselves autonomously such that a globally correctly coloured graph results. Such a dynamic is precisely the graph colouring dynamics mentioned above, and investigations into various heuristics with which the nodes can be equipped already exist. Tracking the dynamics of these heuristics, one notices a drastic difference from global algorithms: in contrast to the global algorithm, “ordering avalanches” begin at various locations simultaneously. The simple reason is that a node at position 1 cannot detect whether an avalanche has already started at the distant position 2.

As a result, however, such avalanches frequently lead to locally valid solutions that are globally incompatible. Whereas at one location colours are distributed in the sequence ABAB, at another location the sequence BABA holds (depicted in Figure 5). This phenomenon of locally correct and globally incompatible solutions can be compared with filter bubbles: within each bubble, an opinion on a topic exists that is, from the perspective of its members, entirely logical. Globally, however, the opinions of the bubbles are incompatible.

By making small changes to the underlying graph, it is now possible to investigate the extent to which the topology supports or prevents the stabilisation of locally correct and globally incompatible solutions. It turns out that the presence of several clusters — that is, different locations in the network that are very strongly interconnected among themselves — rapidly leads to valid and stable local solutions. These respective local solutions can, however, differ globally, which makes finding a global solution more difficult. In contrast, long-range connections synchronise the solution regimes during their formation. Accordingly, globally incompatible solutions are avoided and a global solution is reached more quickly.

Outlook

At the beginning of this text, starting from observable subjectivity, we established the basic assumptions of polycontextural logic. We then introduced the proömial relation and used it to explain how the generation of new knowledge — or the insight into one’s own subjectivity — can be accounted for. These processes were illustrated using the example of wave–particle duality.

How can we now use polycontextural logic to improve the capabilities of AI? To answer this question, we examined two models more closely, which convey fundamental knowledge about PCL. In the first model, various agents strive for conformity. Since the agents operate according to different logics, they cannot reach agreement. Only through the introduction of a simple form of self-reflection can the agents synchronise their logics over time and achieve conformity. Using this model, we were therefore able to confirm that the basic assumptions of PCL enable the emergence of a consistent solution — even, and especially, when the interacting instances initially follow different logics.

In the second model, we investigated the interplay between network topology and the synchronisation of solutions. We were able to show that long-range connections support the emergence of a global solution. Short-range connections, in contrast, accelerate local agreement and stabilise the resulting local solutions against external influences.

With the insights gained from these preliminary investigations, we are now in a position to transfer the mechanisms of polycontextural logic to AI. The next logical step is therefore to connect various AI systems via a network. These different AIs are then to be given the ability to recognise their own subjectivity — in accordance with the proömial relation — and to learn accordingly from this. Based on our preliminary work, we already know which fundamental mechanisms within each AI can enable self-reflection. Additionally, we can already make statements about how the individual AIs are likely to need to be connected in order to generate locally or globally valid knowledge.

Frequently Asked Questions

What is polycontextural logic?

Polycontextural logic (PCL), developed by Gotthard Günther, splits one all-encompassing logic into many interconnected contextures. Classical logic holds within each contexture, but the values “true” and “false” can differ between them. This formal split is what allows the framework to represent subjectivity and self-reference without collapsing into contradiction.

What is the proömial relation?

The proömial relation is Günther’s basic building block of self-reflection. A third subject C observes how two others, A and B, form their opinions, rather than judging the object itself. Seeing that A and B can both be correct yet disagree, C learns that distinct, jointly valid standpoints exist — the seed of reflecting on its own perspective.

Why can’t classical AI achieve self-reflection?

Today’s AI is “weak” and mono-contextural: it assumes a single universal logic with globally fixed truth values. Because it cannot step outside that one logic, it cannot de-objectify its own perspective. Self-reflection requires comparing one’s standpoint against others, which a single global logic structurally forbids.

What do the network models show?

Two simulations make PCL concrete. In the consensus model, agents with different internal “colour” logics synchronise only once a simple self-reflection rule is added. In graph-colouring dynamics, network topology decides the outcome: long-range links promote a global solution, while tight clusters stabilise locally valid but globally incompatible states, a model of filter bubbles.

References

• Falk, J., Eichler, E., Windt, K., Hütt, M.-T., 2022. Collective patterns and stable misunderstandings in networks striving for consensus without a common value system. Sci. Rep. 12, 3028. https://doi.org/10.1038/s41598-022-06880-7
• Falk, J., Eichler, E., Windt, K., Hütt, M.-T., 2021. Physics is Organized Around Transformations Connecting Contextures in a Polycontextural World. Found. Sci. https://doi.org/10.1007/s10699-021-09814-0
• Günther, G., 1979. Beiträge zur Grundlegung einer operationsfähigen Dialektik / Beiträge zur Grundlegung einer operationsfähigen Dialektik (2): Reflexion – Logische … natürlichen Zahl – Dialektischer Materialismus, Unveränderter Print-on-Demand-Nachdruck der Ausgabe von 1979. ed. Felix Meiner Verlag, Hamburg.
• Günther, G., 1976. Beiträge zur Grundlegung einer operationsfähigen Dialektik: Metakritik der Logik, nicht-aristotelische Logik, Reflexion, Stellenwerttheorie, Dialektik, Cybernetic ontology, Morphogrammatik, transklassische Maschinentheorie. Felix Meiner Verlag.
• Hadzhiev, B., Windt, K., Bergholz, W., Hütt, M.-T., 2009. A model of graph coloring dynamics with attention waves and strategic waiting. Adv. Complex Syst. 12, 549–564. https://doi.org/10.1142/S0219525909002386